Module Amenability for Semigroup Algebras

Abstract

We extend the concept of amenability of a Banach algebra A to the case that there is an extra A -module structure on A, and show that when S is an inverse semigroup with subsemigroup E of idempotents, then A = ℓ 1 (S) as a Banach module over A = ℓ 1 (E) is module amenable iff S is amenable. When S is a discrete group, ℓ 1 (E) = C and this is just the celebrated Johnson's theorem.1991 Mathematics Subject Classification. Primary 43A07: Secondary 46H25.

A generalization of the $$n$$ n -weak module amenability of banach algebras

“…Hence l 1 (S) is a Banach algebra and a Banach l 1 (E)-module with compatible actions [1]. Here we let l 1 (E) act on l 1 (S) by multiplication from right and trivially from left, that is: δ e · δ s = δ s , δ s · δ e = δ se = δ s * δ e (e ∈ E, s ∈ S), (4.1)…”

“…The notion of module amenability was introduced by Amini [1] for a class of Banach algebras that are modules over another Banach algebra with compatible actions, and he showed that for an inverse semigroup S with the set of idempotents E, the semigroup algebra l 1 (S) is l 1 (E)-module amenable if and only if S is amenable. Later, A. Bodaghi generalized this concept of amenability in [7] by using module homomorphisms.…”

Weak module amenability of triangular Banach algebras

“…Abdolrasoul Pourabbas -Ebrahim Nasrabadi Amini in [1] developed the concept of module amenability for a class of Banach algebras which is in fact a generalization of the Johnson's amenability. For example for every inverse semigroup S with subsemigroup E of idempotents, he showed that the 1 (E)-module amenability of 1 (S) is equivalent to amenability of S. Duncan and Namioka in [4] for E-unitary semigroup S have shown that 1 (S) is not amenable, whenever E = E S , the set of idempotent elements in S, is finite.…”

Weak amenability for the second dual of Banach modules

“…In other words, all authors investigated and obtained some results related to amenability of Banach algebras while many Banach algebras can be considered as a Banach module over another Banach algebras. Amini [1] used this fact and developed the concept of module amenability for a Banach algebra A to the case that there is an extra A-module structure on A. He showed that for an inverse semigroup S with the set of idempotents E, l 1 .S / is l 1 .E/-module amenable if and only if S is amenable.…”